Cycle Counting under Local Differential Privacy for Degeneracy-bounded Graphs
Quentin Hillebrand, Vorapong Suppakitpaisarn, Tetsuo Shibuya
TL;DR
This work addresses cycle counting under edge local differential privacy for graphs with bounded degeneracy. It introduces a private vertex-ordering preprocessing step based on noisy degree estimates, followed by triangle and odd-cycle counting using a combination of unbiased randomized response and Laplacian queries with restricted sensitivity, achieving favorable error bounds. The main results show an expected ℓ2-error of $O\left(\delta^{1.5} n^{0.5} + \delta^{0.5} d_{\max}^{0.5} n^{0.5}\right)$, reducing to $O(n)$ when $\delta=Θ(1)$, and extending to cycles of length $k$ with $O(n^{(k-1)/2})$ error for degeneracy-bounded graphs. This yields the first purely local DP method with meaningful accuracy for counting cycles longer than triangles in such graph classes, supporting practical private graph analytics on social-network-like data.
Abstract
We propose an algorithm for counting the number of cycles under local differential privacy for degeneracy-bounded input graphs. Numerous studies have focused on counting the number of triangles under the privacy notion, demonstrating that the expected $\ell_2$-error of these algorithms is $Ω(n^{1.5})$, where $n$ is the number of nodes in the graph. When parameterized by the number of cycles of length four ($C_4$), the best existing triangle counting algorithm has an error of $O(n^{1.5} + \sqrt{C_4}) = O(n^2)$. In this paper, we introduce an algorithm with an expected $\ell_2$-error of $O(δ^{1.5} n^{0.5} + δ^{0.5} d_{\max}^{0.5} n^{0.5})$, where $δ$ is the degeneracy and $d_{\max}$ is the maximum degree of the graph. For degeneracy-bounded graphs ($δ\in Θ(1)$) commonly found in practical social networks, our algorithm achieves an expected $\ell_2$-error of $O(d_{\max}^{0.5} n^{0.5}) = O(n)$. Our algorithm's core idea is a precise count of triangles following a preprocessing step that approximately sorts the degree of all nodes. This approach can be extended to approximate the number of cycles of length $k$, maintaining a similar $\ell_2$-error, namely $O(δ^{(k-2)/2} d_{\max}^{0.5} n^{(k-2)/2} + δ^{k/2} n^{(k-2)/2})$ or $O(d_{\max}^{0.5} n^{(k-2)/2}) = O(n^{(k-1)/2})$ for degeneracy-bounded graphs.